Factoring of positive definite functions on semigroups

Semiperfect finitely generated abelian semigroups without involution

“…This we can do already. We note that the last Theorem in [12] is a partial solution of the factoring problem for semigroups S of class M , which is complete in case S is finitely generated. Thus, it suffices to show that every semiperfect finitely generated semigroup is of class M .…”

“…We shall show that in order for a finitely generated abelian semigroup S with arbitrary involution to be semiperfect it is necessary that S be of class M . The factoring problem on semigroups of class M was considered in [12]. The main result states that if S is a semigroup of class M satisfying S = S + S then every positive semidefinite function on S factors via χ .…”

“…This is a straightforward generalization of [4], Theorem 2, which states the same for semigroups with zero. The main result of [12] is not quite enough for our purposes since a semiperfect finitely generated semigroup with the identical involution does not necessarily satisfy S = S + S. Fortunately, in [12] we also considered the case that S is of class M but S = S + S. The second main result of [12] which is a union of equivalence classes with respect to ∼ and which is itself an equivalence relation on some subset of E, there should exist (e, f ) ∈ A such that e + f * ∈ A + S. The condition is sufficient if the set E is finite (in particular, if S is finitely generated). This is the condition that goes into the characterization of semiperfect (or equivalently, completely semiperfect) finitely generated abelian semigroups considered with the identical involution, in addition to the conditions necessary and sufficient for semiperfectness of χ(S).…”

“…The conditions is equivalent to each * -archimedean component of S being cancellative [12]. It is easy to see that every C-separative semigroup is * -separative.…”

Stieltjes Perfect Semigroups are Perfect

“…For a * -semigroup S without zero, positive definite functions on S do not in general factor via χ. For a sufficient condition, see [10].…”